卡斯帕尔·韦塞尔Caspar Wessel

卡斯帕尔·韦塞尔的父亲Jonas Wessel和他的祖父都是教会牧师。他的母亲Helene Marie Schumacher与Jonas育有一个大家庭，共有十四个孩子；韦塞尔是十四个孩子中的第六个。韦塞尔与他的两个哥哥Johan Herman Wessel和Ole Christopher Wessel一起，从1757年到1763年在克里斯蒂安尼亚市的克里斯蒂安尼亚大教堂学校就读（克里斯蒂安尼亚后来改名为克里斯蒂安尼亚，1925年又改名为奥斯陆）。

韦塞尔无法在挪威上大学，因为当时挪威没有大学。由于挪威与丹麦联合，挪威人接受大学教育的自然去处是丹麦，而韦塞尔的兄弟Johan Herman和Ole Christopher已于1761年进入哥本哈根大学。韦塞尔从1763年到1764年在哥本哈根大学学习了一年，但可以想象，这个大家庭给他父母的财务状况带来了相当大的压力。正如我们将在下文看到的，这导致韦塞尔和他的兄弟Ole Christopher都在丹麦皇家科学院获得了测量员的职位。

Ole Christopher 和 韦塞尔 在哥本哈根大学攻读法律学位。Ole Christopher 开始做测量员以帮助支付大学费用，而 韦塞尔 还在上学。Ole Christopher 于1770年获得法律学位，随后在挪威法律界达到了非常高的地位。Johan Herman（比 韦塞尔 大三岁）成为了一名诗人，并且是 韦塞尔 的孩子中唯一在 Encyclopaedia Britannica 中有自己条目的人（尽管 韦塞尔 在三篇数学文章中被提及）。Johan Herman 被描述为：-

……作家和才子，以其警句和轻诗以及对新古典主义悲剧的著名戏仿而闻名。

Royal Danish Academy 开始了一个雄心勃勃的项目，对丹麦进行地形测量，并利用三角测量确定地理坐标。该项目由专业测量员 Thomas Bugge 和哥本哈根数学教授 Christen Hee 领导。Ole Christopher 从1762年起受雇为该项目的一名测量员，当他在1764年需要一名助手时，他的兄弟 韦塞尔 加入项目来帮助他。韦塞尔 和他的兄弟 Ole Christopher 一样，继续攻读法律学位，最终在十五年后获得。然而到那时，他已如此投入测量工作，以至于他在余下的工作生涯中都留在了这个行业。

韦塞尔 一生都遭受经济困难。当然，他作为助手挣得太少，以至于他请求允许他绘制地图以及进行测量，以便他的收入足以让他生存。这一请求得到了批准，他被赋予了相当多的增加的责任，根据三角测量调查收集的数据绘制地图。他绘制的地图标志着[4]：-

……城镇、教堂、城堡、磨坊和树林的位置，道路与溪流的走向，以及海岸线和岛屿的方位。

原本打算为韦塞尔完成大学学业提供经济支持的工作，已经变成了一项如此重大的任务，以至于他没有足够的时间来学习。他担心在科学院如此高强度的工作下永远无法完成学位，但又知道没有这份收入就无法维持生计，于是请求带薪休假一年以完成学位课程。数学教授Christen Hee大力支持韦塞尔的申请，写道（例如见[4]）：-

在所有测量员中，没有人比[韦塞尔]对我们更有用。在夏季他从事测量，冬季则担任绘图员，这十四年来他一直从事测量工作，损害了他的健康，并妨碍了他的学业，以至于如果他再次中断学业，他就完了，再也无法重拾学业。去年冬天，当他半情愿半不情愿地不得不绘制西兰岛的总图时，他的学业再次受到干扰，当时我向他承诺再也不打扰他的研究。

韦塞尔的休假获得了批准，他确实得以完成法律学位。然而，休假一年后，他又回到了测量和地图绘制的工作中。这类工作需要高深的数学技能，而韦塞尔是寻找新方法和技术的创新者。当他编写工作报告时，韦塞尔附上了简短的文章，解释他所采用方法背后的理论思想。

1782年5月，韦塞尔从丹麦皇家科学院的工作中获释，以便对奥尔登堡公国进行三角测量。奥尔登堡于1667年归丹麦控制，但在1773年，即韦塞尔被要求对其进行测量前不久，丹麦的克里斯蒂安七世用奥尔登堡交换了荷尔斯泰因-戈托普。领导丹麦皇家科学院测量工作的Bugge写了一封信推荐韦塞尔担任该职位（例如见[4]）：-

他拥有大量关于代数、三角学和数学几何的理论知识，就最后一点而言，他针对地理测量中最困难的问题提出了一些新颖而优美的解法。

韦塞尔在1785年夏天之前一直从事奥尔登堡的测量工作，之后他回到丹麦皇家科学院的工作中。他一直在发展越来越精密的测量数学方法，并在1787年写的一份报告中充分解释了这些方法。这份报告已经包含了韦塞尔的杰出数学创新，即复数的几何解释。

到1796年，韦塞尔完成了丹麦的三角测量，并利用这些数据制作了该国第一张真正精确的地图。同年，他写了他唯一的一篇数学论文，并于1797年3月10日提交给丹麦皇家科学院的一次会议。直到前一年，科学院才放宽了所有论文必须由科学院成员撰写的规定，而韦塞尔的论文是第一篇被接受的非成员撰写的论文。

韦塞尔作为数学家的名声完全建立在这篇论文上，该论文于1799年发表，首次给出了复数的几何解释。今天我们称这种几何解释为让-罗贝尔·阿尔冈图，但韦塞尔的工作更早。它在1806年被让-罗贝尔·阿尔冈重新发现，1831年又被卡尔·弗里德里希·高斯重新发现。（值得注意的是，卡尔·弗里德里希·高斯重做了韦塞尔工作的另一部分，因为他在1824年左右重新对奥尔登堡进行了三角测量。）

当然，将复数的几何解释称为让-罗贝尔·阿尔冈图并非不合理，因为正是让-罗贝尔·阿尔冈的工作具有影响力。在数学界得知韦塞尔更早的发表之前，它就被如此命名了。事实上，韦塞尔的论文直到1895年才被数学界注意到，当时Juel提请注意它，同年，索菲斯·李重新发表了韦塞尔的论文。由Zeuthen翻译的法文译本于1897年出版，直到更晚才有英文版：[7]。

提腾比尔的提腾斯是18世纪末哥本哈根大学的数学和哲学教授，正是由于他的鼓励，韦塞尔才向科学院提交了他的论文。事实上，韦塞尔不是科学院成员，当他的论文被宣读时他并不在场。是提腾比尔的提腾斯于1797年3月10日向科学院提交了这篇论文。人们只能推测，尽管提腾比尔的提腾斯鼓励韦塞尔，他不可能意识到其重要性，否则他肯定会将其从丹麦语翻译成德语，从而确保韦塞尔获得作为数学家的世界性声誉，而这一声誉直到最近才得以实现。

我们称韦塞尔的工作为非凡的，事实上尽管功劳归于让-罗贝尔·阿尔冈，许多数学史家认为韦塞尔的贡献是[1]：-

……在精神上优于让-罗贝尔·阿尔冈的，并且更为现代。

在[1]文章中，让-罗贝尔·阿尔冈和韦塞尔的方法被比较和对照。当然韦塞尔是一名测量员，他的论文是由他的测量和制图工作所激发的：-

韦塞尔的发展相当直接地从几何问题出发，经过几何直观推理，到达一个代数公式。让-罗贝尔·阿尔冈从代数量开始，并为它们寻求几何表示。……韦塞尔的初始表述非常清晰、直接、简洁且现代。遗憾的是，它近一个世纪都未被赏识，因此没有产生它应得的影响。

然而，对于韦塞尔的唯一一篇数学论文，人们所声称的不仅仅是复数的第一个几何解释。在[3]中，Crowe将第一个添加向量的人归功于韦塞尔。这再次显示了韦塞尔思想的深度，但同样，由于这篇论文未被注意，它对数学发展没有影响，尽管它发表在Mémoires of the Royal Danish Academy上，而无论以何种标准衡量，那都是出版物的一个主要来源。

在许多方面，韦塞尔都是一个非凡的人，这里我们不仅指他的数学才华。尽管贫穷，他拒绝接受受委托绘制地图的报酬。他确实接受了丹麦皇家科学院因他在地图方面的工作而颁发的银质奖章，但奖章并没有使他富裕起来。1805年，他60岁时辞去了在丹麦皇家科学院的工作。然而，他并没有放弃绘制地图，必要时他仍然帮助丹麦皇家科学院。他退休后绘制的地图之一是一幅石勒苏益格-荷尔斯泰因地图，这是拿破仑·波拿巴要求的。

1815年，韦塞尔被授予丹尼布洛骑士团骑士称号：-

……以表彰他在测量学方面的杰出贡献。

Caspar Wessel's father, Jonas Wessel, and his grandfather were both ministers in the church. His mother, Helene Marie Schumacher and Jonas had a large family consisting of fourteen children; Caspar was the sixth of the fourteen. Caspar, together with his two elder brothers Johan Herman Wessel and Ole Christopher Wessel, attended Christiania Cathedral School in the city of Christiania from 1757 until 1763 (Christiania was later renamed Kristiania, then Oslo in 1925).

Caspar could not attend university in Norway for, at that time, there were no Norwegian universities. Since Norway was united with Denmark, the natural place for Norwegians to go for a university education was Denmark, and Caspar's brothers Johan Herman and Ole Christopher were already at the University of Copenhagen, having entered in 1761. Caspar spent one year, from 1763 to 1764, studying at the University of Copenhagen but, as one might imagine, the large family was putting quite a strain on his parent's finances. As we shall see below, this led to both Caspar and his brother Ole Christopher obtaining positions as surveyors with the Royal Danish Academy of Sciences.

Ole Christopher and Caspar were studying at the University of Copenhagen for a law degree. Ole Christopher started to work as a surveyor to help pay his way through university while Caspar was still at school. Obtaining his law degree in 1770, Ole Christopher went on to reach a very elevated position in the legal profession in Norway. Johan Herman (three years older than Caspar) became a poet and is the only one of the Wessel children to have their own entry in Encyclopaedia Britannica (although Caspar is mentioned in three of the mathematics articles). Johan Herman is described as a:-

... writer and wit, known for his epigrams and light verse and for a famous parody of neoclassical tragedy.

The Royal Danish Academy began an ambitious project to undertake a topographical survey of Denmark and also to use triangulation to determine geographical coordinates. The project was led by Thomas Bugge, a professional surveyor, and Christen Hee, the professor of mathematics in Copenhagen. Ole Christopher was employed as a surveyor on this project from 1762, and, when he needed an assistant in 1764, his brother Caspar joined the project to help him. Caspar, like his brother Ole Christopher, continued to study for his law degree which he eventually achieved after fifteen years. However by that time he was so involved with surveying that he remaining in that profession the rest of his working life.

Throughout his life Wessel suffered financial hardship. Certainly he earned to little as an assistant that he requested he might be allowed to draw maps, as well as surveying, so that his income might be sufficient to allow him to survive. This request was granted and he was given quite a lot of increased responsibility drawing maps based on the data which was being gathered from the triangulation survey. The maps he drew marked [4]:-

... the locations of towns, churches, castles, mills, and woods, the courses of roads and streams, and the positions of coastlines and islands.

Work which had originally been intended to provide Wessel with the financial support to complete his university course had become such a major undertaking that there was not enough time left for him to study. Fearing that he would never complete his degree with his work for the Academy running at this high level, yet knowing that he could not survive financially without this income, he requested a sabbatical year on full pay to complete his degree course. Christen Hee, the professor of mathematics, strongly supported Wessel's application writing (see for example [4]):-

None of the surveyors has been more useful to us than [Wessel] has, during the summers he has been surveying and in the winter time he has been working as a designator, which in the fourteen years he has stayed with the surveying has ruined his health and been an obstacle to his studies in such a way that if he once again has to interrupt his studies he is lost and will never pick them up again. Last winter when he half unwillingly, half willingly had to draw the general map of Zealand he was once more distracted in his studies, and then I promised him never again to disturb his circles.

Wessel's sabbatical was granted and he was indeed able to complete his law degree. However, after the sabbatical year he returned to his tasks of surveying and map construction. Such tasks required demanding mathematical skills and Wessel was an innovator in finding new methods and techniques. When he compiled reports on his work, Wessel appended short articles explaining the theoretical ideas behind the methods he was employing.

In May 1782 Wessel was released from his work with the Royal Danish Academy so that he could conduct a trigonometrical survey of the duchy of Oldenburg. Oldenburg had come under Danish control in 1667 but in 1773, not long before Wessel was asked to survey it, Oldenburg had been exchanged by Christian VII of Denmark for Holstein-Gottorp. Bugge, who headed the survey work at the Royal Danish Academy wrote a letter recommending Wessel for the post (see for example [4]):-

He possesses a lot of theoretical knowledge of algebra, trigonometry and mathematical geometry, and as far as the last point is concerned, he has come up with some new and beautiful solutions to the most difficult problems in geographical surveying.

Wessel worked on the survey of Oldenburg until the summer of 1785 when he returned to his work with the Royal Danish Academy. He had been developing more and more sophisticated mathematical methods of surveying and these he explained fully in a report he wrote in 1787. This report already contains Wessel's brilliant mathematical innovation, namely the geometric interpretation of complex numbers.

By 1796 Wessel had completed the triangulation of Denmark and used the data to produce the first really accurate map of the country. In the same year he wrote his one and only mathematical paper and presented it to a meeting of the Royal Danish Academy on 10 March 1797. Only in the year before had the Academy relaxed its rule that all papers must be written by members of the Academy, and Wessel's paper was the first to be accepted which was not authored by a member.

Wessel's fame as a mathematician rests solely on this paper, which was published in 1799, giving for the first time a geometrical interpretation of complex numbers. Today we call this geometric interpretation the Argand diagram but Wessel's work came first. It was rediscovered by Argand in 1806 and again by Gauss in 1831. (It is worth noting that Gauss redid another part of Wessel's work, for he retriangulated Oldenburg in around 1824.)

Of course it is not unreasonable to call the geometrical interpretation of complex numbers the Argand diagram since it was Argand's work which was influential. It was so named before the world of mathematics learnt of Wessel's prior publication. In fact Wessel's paper was not noticed by the mathematical community until 1895 when Juel draw attention to it and, in the same year, Sophus Lie republished Wessel's paper. A French translation, made by Zeuthen, was published in 1897 and it was not available in English until even later: [7].

Johannes Nikolaus Tetens was a professor of mathematics and philosophy at the University of Copenhagen near the end of the 18th century, and it was because of his encouragement that Wessel presented his paper to the Academy. In fact Wessel, not being a member of the Academy, was not present when his paper was read. It was Tetens who presented the paper to the Academy on 10 March 1797. One can only suppose that, despite Tetens encouraging Wessel, he could not have realised its importance for otherwise he certainly could have translated it from Danish to German and thus ensured for Wessel the world-wide fame as a mathematician which he has not achieved until very recent times.

We have called Wessel's work remarkable, and indeed although the credit has gone to Argand, many historians of mathematics feel that Wessel's contribution was [1]:-

... superior to and more modern in spirit to Argand's.

In the [1] article the approaches by Argand and Wessel are compared and contrasted. Of course Wessel was a surveyor and his paper was motivated by his surveying and cartography work:-

Wessel's development proceeded rather directly from geometric problems, through geometric-intuitive reasoning, to an algebraic formula. Argand began with algebraic quantities and sought a geometric representation for them. ... Wessel's initial formulation was remarkably clear, direct, concise and modern. It is regrettable that it was not appreciated for nearly a century and hence did not have the influence it merited.

However more is claimed for Wessel's single mathematical paper than the first geometric interpretation of complex numbers. In [3] Crowe credits Wessel with being the first person to add vectors. Again this shows the depth of Wessel's thinking but again, as the paper was unnoticed it had no influence on mathematical development despite appearing in the Mémoires of the Royal Danish Academy which by any standard was a major source of publications.

In many ways Wessel was a remarkable person, and here we are not only referring to his mathematical brilliance. Despite his poverty, he refused to accept payment for maps which he had been commissioned to make. He did accept the award of a silver medal from the Royal Danish Academy for his work on maps but medals did not make him well off. He resigned from his work with the Royal Danish Academy in 1805 when he was 60 years old. He did not give up drawing maps, however, and he still helped the Royal Danish Academy when necessary. One of the maps he drew after he retired was a map of Schleswig-Holstein which had been requested by Napoleon Bonaparte.

In 1815 Wessel was made a Knight of the Order of Dannebrog:-

... in recognition of his exceptional contribution to surveying.